Imaging by wavefield extrapolation

Imaging by wavefield extrapolation (WE) is based on recursive continuation of the wavefields $\u$ from a given depth level to the next by means of an extrapolation operator ${\bf E}$:  

$$\u_{z+\Delta z}= {\bf E}_z\left[\u_{z } \right].$$ (1)

This recursive relation can also be explicitly written in matrix form as

$$\begin{eqnarray} \left(\matrix { {{\bf 1}} & {\bf 0}& {\bf 0}&\cdots& {\bf 0}& ... ...\matrix{ \d_0 \cr 0 \cr 0 \cr\vdots\cr 0 \cr } \right), \nonumber \end{eqnarray}$$

or in a more compact notation as:

$$\left({\bf 1}- {\bf E}\right)\u = \d,$$ (2)

where the vector $\d$ stands for the recorded data, $\u$ for the extrapolated wavefield, ${\bf E}$ for the extrapolation operator and ${\bf 1}$ for the identity operator.

The wavefield at every depth level $\u_{z }$ is imaged using an imaging operator ${\bf I}_z$:

$$\r_{z }= {\bf I}_z\left[\u_{z } \right],$$ (3)

where $\r_{z }$ stands for the image at some depth level. We can write the same relation in compact matrix form as:

$$\r = {\bf I}{\u},$$ (4)

where $\r$ stands for the image, and ${\bf I}$ stands for the imaging operator which is applied to the extrapolated wavefield $\u$.